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Theorem bnj873 31541
Description: Technical lemma for bnj69 31625. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj873.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj873.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj873.8 (𝜓′[𝑔 / 𝑓]𝜓)
Assertion
Ref Expression
bnj873 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfv 2015 . . 3 𝑔𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
3 nfcv 2970 . . . 4 𝑓𝐷
4 nfv 2015 . . . . 5 𝑓 𝑔 Fn 𝑛
5 bnj873.7 . . . . . 6 (𝜑′[𝑔 / 𝑓]𝜑)
6 nfsbc1v 3683 . . . . . 6 𝑓[𝑔 / 𝑓]𝜑
75, 6nfxfr 1954 . . . . 5 𝑓𝜑′
8 bnj873.8 . . . . . 6 (𝜓′[𝑔 / 𝑓]𝜓)
9 nfsbc1v 3683 . . . . . 6 𝑓[𝑔 / 𝑓]𝜓
108, 9nfxfr 1954 . . . . 5 𝑓𝜓′
114, 7, 10nf3an 2006 . . . 4 𝑓(𝑔 Fn 𝑛𝜑′𝜓′)
123, 11nfrex 3216 . . 3 𝑓𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)
13 fneq1 6213 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
14 sbceq1a 3674 . . . . . 6 (𝑓 = 𝑔 → (𝜑[𝑔 / 𝑓]𝜑))
1514, 5syl6bbr 281 . . . . 5 (𝑓 = 𝑔 → (𝜑𝜑′))
16 sbceq1a 3674 . . . . . 6 (𝑓 = 𝑔 → (𝜓[𝑔 / 𝑓]𝜓))
1716, 8syl6bbr 281 . . . . 5 (𝑓 = 𝑔 → (𝜓𝜓′))
1813, 15, 173anbi123d 1566 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′)))
1918rexbidv 3263 . . 3 (𝑓 = 𝑔 → (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)))
202, 12, 19cbvab 2952 . 2 {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)} = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
211, 20eqtri 2850 1 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1113   = wceq 1658  {cab 2812  wrex 3119  [wsbc 3663   Fn wfn 6119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-fun 6126  df-fn 6127
This theorem is referenced by:  bnj849  31542  bnj893  31545
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